How can implement EM-GMM in python?

Question

I have implemented EM algorithm for GMM using this post GMMs and Maximum Likelihood Optimization Using NumPy unsuccessfully as follows:

import numpy as np

de         


        

Answer 1

As I mentioned in the comment, the critical point that I see is the means initialization. Following the default implementation of sklearn Gaussian Mixture, instead of random initialization, I switched to KMeans.

import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
plt.style.use('seaborn')

eps=1e-8 

def PDF(data, means, variances):
    return 1/(np.sqrt(2 * np.pi * variances) + eps) * np.exp(-1/2 * (np.square(data - means) / (variances + eps)))

def EM_GMM(data, k=3, iterations=100, init_strategy='kmeans'):
    weights = np.ones((k, 1)) / k # shape=(k, 1)
    
    if init_strategy=='kmeans':
        from sklearn.cluster import KMeans
        
        km = KMeans(k).fit(data[:, None])
        means = km.cluster_centers_ # shape=(k, 1)
        
    else: # init_strategy=='random'
        means = np.random.choice(data, k)[:, np.newaxis] # shape=(k, 1)
    
    variances = np.random.random_sample(size=k)[:, np.newaxis] # shape=(k, 1)

    data = np.repeat(data[np.newaxis, :], k, 0) # shape=(k, n)

    for step in range(iterations):
        # Expectation step
        likelihood = PDF(data, means, np.sqrt(variances)) # shape=(k, n)

        # Maximization step
        b = likelihood * weights # shape=(k, n)
        b /= np.sum(b, axis=1)[:, np.newaxis] + eps

        # updage means, variances, and weights
        means = np.sum(b * data, axis=1)[:, np.newaxis] / (np.sum(b, axis=1)[:, np.newaxis] + eps)
        variances = np.sum(b * np.square(data - means), axis=1)[:, np.newaxis] / (np.sum(b, axis=1)[:, np.newaxis] + eps)
        weights = np.mean(b, axis=1)[:, np.newaxis]
        
    return means, variances

This seems to yield the desired output much more consistently:

s = np.array([25.31      , 24.31      , 24.12      , 43.46      , 41.48666667,
              41.48666667, 37.54      , 41.175     , 44.81      , 44.44571429,
              44.44571429, 44.44571429, 44.44571429, 44.44571429, 44.44571429,
              44.44571429, 44.44571429, 44.44571429, 44.44571429, 44.44571429,
              44.44571429, 44.44571429, 39.71      , 26.69      , 34.15      ,
              24.94      , 24.75      , 24.56      , 24.38      , 35.25      ,
              44.62      , 44.94      , 44.815     , 44.69      , 42.31      ,
              40.81      , 44.38      , 44.56      , 44.44      , 44.25      ,
              43.66666667, 43.66666667, 43.66666667, 43.66666667, 43.66666667,
              40.75      , 32.31      , 36.08      , 30.135     , 24.19      ])
k=3
n_iter=100

means, variances = EM_GMM(s, k, n_iter)
print(means,variances)
[[44.42596231]
 [24.509301  ]
 [35.4137508 ]] 
[[0.07568723]
 [0.10583743]
 [0.52125856]]

# Plotting the results
colors = ['green', 'red', 'blue', 'yellow']
bins = np.linspace(np.min(s)-2, np.max(s)+2, 100)

plt.figure(figsize=(10,7))
plt.xlabel('$x$')
plt.ylabel('pdf')

sns.scatterplot(s, [0.05] * len(s), color='navy', s=40, marker=2, label='Series data')

for i, (m, v) in enumerate(zip(means, variances)):
    sns.lineplot(bins, PDF(bins, m, v), color=colors[i], label=f'Cluster {i+1}')

plt.legend()
plt.plot()

Finally we can see that the purely random initialization generates different results; let's see the resulting means:

for _ in range(5):
    print(EM_GMM(s, k, n_iter, init_strategy='random')[0], '\n')

[[44.42596231]
 [44.42596231]
 [44.42596231]]

[[44.42596231]
 [24.509301  ]
 [30.1349997 ]]

[[44.42596231]
 [35.4137508 ]
 [44.42596231]]

[[44.42596231]
 [30.1349997 ]
 [44.42596231]]

[[44.42596231]
 [44.42596231]
 [44.42596231]]

One can see how different these results are, in some cases the resulting means is constant, meaning that inizalization chose 3 similar values and didn't change much while iterating. Adding some print statements inside the EM_GMM will clarify that.